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LIBRARY OF CONGRESS. 



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Shelf tH-33. 

UNITED STATES OF AMERICA. 



A PREFATORY ESSAY 

TO THE 

NEW SCIENCE: 



1 

PRECEDED PA r A 



BRIEF RETROSPECTIVE VIEW 

Of Research in the Dojnain of 

GEOMETRY. 



By Chas. De Medici, D. Ph. 



CHICAGO, ILL.; 

A, M. FLANAGAN. 

1S83. 



A PREFATORY ESSAY 

TO THE 

NEW SCIENCE; 



PRECEDED EY A 



BRIEF RETROSPECTIVE VIEW 

Of Research in the Domain of 

GEOMETRY 



By Chas. De MEDicK|^f|«COjJ^ 



CHICAGO, ILL.\_ 

A Hr T-T i AT A ^ A AT^L^ 

A.M. FLANAGAN> 



STrc 



COPYRIGHTED, 1 863, 
BY 

Chas. De Medici. 



v^. 






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!b 



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f \ 



4O* 



ERUATA. 

Page 6. After reading the third line, read the Note on 

the same page. 
Page 23. 24^ below the cut should be 24^. 

Page 27. In the seventh line from the bottom, 15904 
should be 135904. 

Page 28. In the middle of the page 42470 — 2 should 
read 42470-^2. 

Page 29. In line seventeen from the top place a deci- 
mal point after the figure 5. 

These errors will not appear in the larger work. 



RETROSPECTIVE VIEW 



IN THE DOMAIN OF 



GEOMETRY 



Whatever was done, previous to the Grecian era of 
intellectual progress, which later led to the solution of 
profound geometric problems, we will leave obscured 
in the twilight of uncertainty. 

With the dawn of Grecian civilization the architec- 
tural art began to develop its beauty and as it grew 
the science of geometry unfolded its hidden power. 

Plato, Euclid, Pythagaros and Procles are names 
allied with the earliest inquiries and discoveries relating 
to the great science. These ancient mathematicians' 
manner of inquiry differed widely from the modern. 
The ancient mathematician was enthusiastic about his 
work. He had no computing tables, no ready made 
calculations, no logarithms, no personal authority to 
lean upon or refer to. To him it was a charming labor 
to roam through this vast labyrinth of inquiry with 
nothing but the square and compasses to guide him. 
Every now and then a crude measurement would 



4 MATHEMATICAL COMMENSURATION. 

furnish a clue to the solution of a difficult problem and 
as often the apparently near approach to a final and 
true result terminated in the discovery of a slight 
error. 

Again and again with unabated energy the search 
would be renewed until an irrevocable law of relation- 
ship was proven or a self-evident fact discovered. 
Thus, each of these earliest toilers added from time 
to time information to the general stock of mathemat- 
ical knowledge. 

In the times we speak of geometry was especially 
applied to architectural designs. Comparatively little 
abstract computation was needed, the square and com- 
passes, the plumb and the level, and a scale of propor- 
tions constituted the principal instruments for calcula- 
tions. 

Later on when the mechanical arts, stimulated by the 
inventive genius of Archimedes, assumed mature propor- 
tions, a new field of inquiry in the domain of geometry was 
opened up. It was then required to compute abstractly 
the relations of mechanical combination. 

Geometry by this time had already become a practical 
science and its fundamental principles, so far as known, 
had been formulated into exact propositions which 
could be proven true by algebraic demonstrations. 

But, to be of practical value to the mechanical arts, 
the algebraic formulas have to be translated into 
numerical expressions, in consequence of which, 
arithmetic was added as an aid, to the science of 
geometry. 



Still later, when Galileo and Kepler announced to 
the world the new theory of a revolving planetary 
system of which the earth we inhabit forms a part, 
then the astronomical field loomed up and presented 
aspects calling for more exact measures. 

Inquiry soon discovered the fact that approximate 
measure, near enough correct to suit the artificial 
mechanism and the architectural designs of man were 
not sufficiently accurate to suit the architecture and 
organism of the universe. 

So far in the history of geometry all attempts to 
establish commensurate relationship between the results 
obtained by algebra and the results obtained by Arith- 
metic had been futile, although many eminent mathe- 
maticians were engaged in the work. This failure gave 
credence to the assumption that geometric quantities 
were of two kinds : commensurable and incommensur- 
able. 

Hence, Mathematicians directed their attention to 
the finding of the nearest approximate results obtainable 
by arithmetic and in the easiest manner. This led to 
the ingeniously devised "logarithms" of Baron Napier, 
which were published in Edinburg in the year 1614. 

The extraordinary facilities with which these numbers 
aided the astronomers in obtaining results nearly true, 
gave to logarithms a virtue which long usage has con- 
verted into apparent necessity. 

Yet, these most valuable factors, the logarithms, are 
constantly improved upon and subject to corrections in 
accordance with errors detected now and then by phe- 



6 MATHEMATICAL COMMENSURATION. 

nomenal events; and most strange, these improvements 
and corrections, instead of bringing the final result 
nearer to the truth seemingly drives it further off. 

Previous to the invention of Logarithms and up to 
the present age, the general theory on which the 
computation of relations between chords and arcs, were, 
and are based, is, that the circumference of a circle is 
composed of points having no dimensions, and that 



Note : This statement may appear bold, but until a 
thoroughly harmonious or commensurate relationship 
shall be established so that every lettered quantity of 
an algebraic formula can be translated into a numerical 
value which will exactly express in finite numbers every 
geometric relation, until then, no one can prove how far 
from, or how near to the truth are the results obtained 
by the best of logarithms when these are applied to ge- 
ometric mensuration ; and were it not that the evidence 
and proof of such a commensurable relationship is pos- 
sible and fully demonstrated in the treatise following 
this preface, the statement would not have been made. 

Neither should it be understood that the author de- 
sires to depreciate the value of logarithms as useful 
means of expediency, but in relation to geometry 
and as a part of that science, logarithms constructed 
on a decimal basis are unscientific and false in founda- 
tion — because : JVo geometric line, divided into ten equal 
parts, will supply a commensurate geometric unit measure. 

On this ground the statement is pertinent to the 
subject, and will be taken up again in its proper place. 



the sides of inscribed and circumscribed polygons could 
by repeatedly bisecting be reduced to infinites- 
imal parts of a line equal to the points of no magni- 
tude. It is only surprising to think that so vague and 
sophistical a theory cculd lead to so near correct results 
as logarithms actually give. 

Archimedes (220 B. C.,) calculated the peripheries 
of the inscribed and described polygons of 96 sides, 
from which calculations he deduced that the circumfer- 
ence of a circle must be between 3$ and 3!?, when the 
diameter is 1. Accordingly, he concluded that the ratio 
7:22, exceeds the truth only to a small extent. This 
estimate comes pretty near the truth, for had he added 
4 2 ! to the number 7 and £ to the number 22, he would 
have had in the final multiples of these fractional num- 
bers the two whole numbers: 289:908; which num- 
bers will be proven by demonstration and on scientific 
ground to be the true ratio. 

Peter Metius, in the 16th century, by means of poly- 
gons of 1536 sides, found that the true ratio was less 
than 4^J and greater than 3^ and so it is, because : 
3!20 =120:377, which is greater than 289:908, and, 3 ,Jjf 6 
= 106:333, which is less than the true ratio 289:908. 
Not being able to find the exact numbers, he took the 
mean difference for the ratio and gave the numbers : 
ii 3 : 355j which are right when we add Jg| to the first, 
and 2 | 9 to the latter. Shortly after, between the years 
1654-1705, James Bernoulli proved by algebraic demon- 
stration that the true ratio between diameter and cir- 



8 MATHEMATICAL COMMENSURATION. 

cumference could be expressed with perfect exactness 
as follows : 

" If the number 4 be divided by 1, 5, 9, 13, and 
every fourth number in succession, and afterwards by 
3, 7, 11, 15, and every fourth number thereafter, the 
difference between the sum of the first set of quotients 
and that of the second, is equal to the ratio." 

The numbers thus obtained and converted into com- 
mensurational values give the true ratio. But when the 
same numbers are converted into decimals, they give an 
approximate ratio only, which in high calculations with- 
out corrections are wide off from the truth. In decimals 
the ratio is written : 3.14159265, etc., infinitely, and is 
the one most commonly used at present.* 

This same problem occupied the attention of Anaxa- 
goras as early as 480 B. C, and it might reasonably be 
inferred from the well-known ratio: 1250:3927=3.1416 
etc., which is of Hindoo origin, that in the far east, 
long before the Grecian era of intellectual ascendency, 
the same problem was studied. 

The failure of so many eminent mathematicians to 
translate the true ratio into exact and finite numbers did 
not for a long time seem to deter others from engaging 
in the same pursuit. 

A Frenchman, Vieta, and a Hollander, Romanus, 



* Later on in this treatise will be given an exposi- 
tion of the problem as it has been treated in the past 
and as it should be treated to make it of practical value 
to science. 



carried the calculations to polygons of more than a 
thousand millions of sides, and in 1590, Ludolph van 
Reuben calculated the circumference of a diameter ex- 
pressed by 35 ciphers and found it within one figure as 
follows: 3i4,i59> 26 5>358,979>3 2 3>846,264,338,327,95o- 
288 or 289? 

This is the so-called Ludolphian number, engraved 
on the tombstone of the discoverer (?) at Leyden ; Hol- 
land. 

A noted authority says : Lagny, in 1719, worked out 
a ratio of 121 decimals ; Dr. Rutherford of Woolwich, 
to 200 figures j Dr. Clauson of Dorpat, to 250 ; Mr. 
Shanks to 607; and he adds — "All these calculations 
agree with Ludolph as far as he goes " — forgetting, no 
doubt, that the farther we go searching for truth in a 
wrong direction, the farther will we be off from it. It 
is most likely that every one of these mathematicians 
took up the calculation where another had left off, tak- 
ing for granted that so far it was right. 

Had either of the followers after Ludolph only 
noticed that a certain odd relationship repeated itself 
whenever the two numbers 288, 289, happened to fall 
together as in the Ludolphian number already given, 
and had they then directed their attention to trace out 
where these two near relatives properly belonged as 
geometric exponents, it is more than likely that long 
ere now the question of mathematical commensuration 
would have been solved. 

Montucla in 1754 published a "History of Research 
for the quadrature of the circle" and in that whole 



IO MATHEMATICAL COMMENSURATION. 

record, which is a vast compilation, there is not one 
distinct method different from another ; all engaged in 
the work seemingly wrought in the old root, and made the 
infinitesimal theory their base of operation. No wonder 
mathematicians failed in their attempt to "square the 
circle" and that circle-squaring was left an idle pastime 
for cranks. 

But it is to be wondered at that so many brilliant 
intellects could be engaged in a work of that kind and 
all were willing to accept the fallacious theory of infini- 
tesimal division and incommensurable relationship as; 
the basis for applied geometry, when they must have 
known that the first fixed principle of all system and 
hence of science, is, a finite and definable limitation. 

Lambert in 1761 and Legendre soon afterward 
pretended to prove that the exact ratio between 
diameter and circumference can not be expressed by 
any number. 

Previous to this unwarrantable statement nearly all 
scientific institutions of learning offered large premiums 
for the discovery of a commensurate ratio between dia- 
meter and circumference of circles, which ratio could 
be expressed and proven true on the one hand by 
algebra and on the other by arithmetic. After the 
announcement made by these supposed best authorities, 
the academy of science at Paris in 1775 and soon after- 
wards the Royal society of London declined to 
examine in future any paper pretending to the quadra- 
ture of the circle. 

In the face of these statistical facts it require some- 



strong evidence and more than ordinary argument to 
even point at a possible way that will lead to more 
scientific results. 

Nothing short of absolute proof by processes easily 
explained and resting on axioms generally recognized, 
will justify presentation of this subject. 

Notwithstanding all that has been said and written 
about the subject, the author respectfully submits a brief 
outline of the manner by which the discovery was 
made and the proof obtained and he hopes that among 
the learned faculties of the United States and in 
Europe there is sufficiently freedom from prejudice to 
have a matter of so vital importance to education fairly 
investigated, when, as in this case, irrefutable proof of 
truth is furnished at the start. 



MATHEMATICAL COMMEXSURATION. 



PREFATORY ESSAY 



NE W SCIENCE : 



MATHEMATICAL COMMENSURATIOX. 



In the course of a life ripe with novel ideas it once 
occurred to the author that the principles of object- 
teaching as applied and practiced in "kindergartens" 
might with great advantage be extended to adult train- 
ing in mastering abstruse branches of knowledge. 

To that end, in testing the practicability of such 
a scheme applied to first principles of geometry, a 
great number of uniform wooden cubes were provided. 

A long row of these cubes were placed side by side. 
Another row of the cubes were formed parallel to the 
first in a manner so that the extremes of their 
diagonals touched each other as it is shown in the 
diagram : 



PREFATORY ESSAY. 



^3 



Side: 17X2/1=408. 




Diagonal: 24x17=408. 



By having several hundreds of these cubes placed in 
succession, it was noticed every 17 diagonals equaled 
in length 24 of the sides, and that, therefore, the ratio 
17 suggested itself. 

This of course by itself was of little account, but it 
was a clue. 

The next obvious experiment was to find a possible 
clue to the ratio of diameter and circumference. 

A number of plastic uniform cubes were formed. 
From some of these a number of globes were made to 
be sure that the same area was contained in each of the 
differently formed quantities. These globes and cubes 
were likewise placed parallel to each other, as shown in. 
the diagram : 



MATHEMATICAL C0MMENSURAT10N. 

Diameter : 28 




Side of Equal Square : 2 2 7 x 4= 908= Circumference. 
289X227=65603 
227X289=65603 

Then it was found that it required 2 89 of the cubes 
to balance in length 227 of the globes, which, when 
reduced to a ratio between a single globe and a single 
cube gave the numbers 227 to the side of an equal 
square, 908 to a circumference and 289 to a diameter. 
This was also but a clue ; yet there was one peculiar 
coincidence with these two relations thus obtained by- 
crude measurements which did not escape observation. 
That peculiarity was the fact that the number 289 
representing the diameter was the square number of 17 
representing the side of a square in the first experiment; 
and because the diameter of a circle apparently is equal 
in length to the side of its described square, here was 
the evidence that some kind of commensurate relation- 
ship existed between the measurements obtained. 

If there was any truth in this apparent fact, a relation 
could also be found between the number 1 7 and the 
side of the equal square. (227.) 



PREFATORY ESSAY. 



*S 



To that end a circle was drawn and a square inscribed 
as shown in the diagram : 




Diagram 1. 

Accordingly 227, the number given for the side of 
the equal square, divided by 1 7 gives the quotient : 1 3^ 
which in its final multiple is commensurable to the 
diameter. 

In order to find the ratio between side of inscribed 
square and side of equal square, and between diagonal 
of inscribed square and diagonal of equal square, the 
ratio \j was applied : as24:i7::i7:i2 24= the side of an in- 
scribed square. 

But as the numbers 13 i T :i2 ^ are of complex denom- 
inations it is preferable to find two numbers more sim- 
ple in expression. To facilitate this finding, a scale of 
numbers was constructed in this wise : 



i6 



MATHEMATICAL COMMENSURATION. 



<* 

CM 

CO 


— 




O 
lO 

M 

cm 



00 

**» 

in 

CO 

M 




cm 

CM 






CM 






o 

CM 






O 







CO 


















to 












CO 
O 








^ 


Sf N 


— 




3 : 




O 


o 








C\ 












CO 


— 













lO 












^t 






tr. 






CM 


~ 




- 


CM 





A line was divided into 1 7 
equal parts, another parallel 
line the same length was di- 
vided into 24 equal parts. 
Each of the i7ths were sub- 
divided into 24 equal parts, 
and each of the 24's were sub- 
divided into 17 equal parts. 
Thus, a scale was obtained 
composed of 408 equal, com- 
mon, and final multiples, of 
which, 17 is the measure for 
each of the 24 sides and 24 
is the measure for each of the 
17 diagonals as it is shown 
in the diagram on this page. 

To prove the construction 
of the scale true, any two 
numbers representing sides 
and diagonals of squares, 
which numbers have been ob- 
tained by the ratio given (17: 
24), shall be cut by one and 
the same number of final mul- 
tiples on the scale — thus : 1 7 
on the left side of the scale 
shall cut the right side of the 
scale at twelve and one-twen- 
ty - fourth, and both these 
lengths shall count the same 



PREFATORY ESSAY. 



17 



number of final multiples because 17X17—289, and 12^ 
X24= 289. So, also, 8*4 on the right side shall cut at 
12 on the left side, because as 17:24:: 8}^: 12. 

f 




Diagram 2. 

By the second trial the radius a c was made equal to 
the number 8)4- Applying the ratio 17:24 the side a b 
of the inscribed square, was found to be 12. 

And because the ratio between diameter and circum- 
ference were expressed by the numbers : 289:908, or as 
the diameter 1 is to the circumference 3^ it follows 
that the ratio between radius and circumference must be 
6.J39 or as 289:1816. 

Hence : 8^x6|| = 53^ which divided by 4 gives the 
side of the equal square d e, the numerical value of 
I 3n- 



1 8 MATHEMATICAL COMMENSURATION. 

But here the puzzle began, because by comparing the 
two diagrams i and 2, it was found that while the num- 
bers and ratios were commensurable to the common rule 
of proportion, the numerical value of the inscribed 
square in its lateral measurements had changed from 
12^ to 12. Was it perhaps because the lines were in- 
verted ? Yes ! and the discovery was made that when 
the side of a square is converted into a diagonal, its 
numerical value is diminished, because of the increased 
magnitude of its areal unit measures, and when a diag- 
onal is converted into a side, its numerical value is in- 
creased because of decrease in the magnitude of its 
areal unit measures. 

Thus, in diagram 1, the line a b is the side of the 
square abed, and measures i2 24 ; in diagram 2, the 
same line ab is the diagonal of the square acb f, and 
measures only 12. 

This discovery led to further investigations which 
disclosed the fact that the integrants composing a root 
or the unit measures of area, do not correspond in their 
length to the linear unit measures composing the length 
of the side of the square ; unless these two diverse 
kind of measures are adjusted to each other by ratio, or 
otherwise happen by accident to be even, such as in the 
triangles : 



PREFATORY ESSAY. 



A full investigation proved that the first proposition 
of the trigonometrical canon commonly used at our insti- 
tutions of learning is not true, because it is based on 
the assumption that the root of a square, numerically 
expressed, is also the true numerical expression for the 
length of the side ; and, that numbers expressing geo- 
metric quantities, such as roots and squares, can be treat- 
ed in computation as if they were abstract numbers. 
Por while it is an admitted truth that the areas of the 
two squares described on the legs of a right angled tri- 
angle, together taken, are equal to the one area describ- 
ed on the hypotenuse of the same triangle, (see prop. 
47, Book I, Euclid,) this axiom only applies to undi- 
vided areas, or to areas composed of differently formed 
integrants, irrespective of any commensurate unit-mea- 
sure. 




This diagram stiows plainly the truth of the prop. 47, 
"but it also shows that no unit measure can be ob- 
tained by the ratio 2:1, as generally applied to in- 
scribed and described squares without change of 



MATHEMATICAL COMMEXSURATION. 



form from a square to an oblong, or by making one 
of the squares fractional in its root without changing 
the number of roots originally contained. 



h A 

A/ 

YY 
YY 

yV 

/V 
/Y 

YY 

YY 
YY 


b\ 


\A 2. 
e \^ 


4- ? 
D/ f 


1 i 1 1 1 i i 1 ! 1 l\ 


/\ i i i i i : i ! i i 



To illustrate, we will suppose that the square abed 
contains 289 square integrants with a root composed of 
17 integrants and the number of root is also 17, since 

289-^-17=17. 

According to the ratio given by the prop. 47, (1:2,) 
the square e f g h shall be composed of 578 square in- 
tegrants, each of the same magnitude as each of the 
289 contained in inscribed square abed. But 24X24= 



PREFATORY ESSAY. 21 

576 does not give the true result, and to adjust this 
difference of 2, in the numerical expression each of the 
areal unit-measures of the described square must -be re- 
duced 2^3 part, because H=2, or each of the areal unit 
measures of the inscribed square must be augmented 
289 part, because Hl—i, which deducted from the number 
289 leaves 288, and 288x2=576. 

It is evident that this adjustment of roots by change 
of numerical expression does not change the actual 
dimension of the square area any more than the subdi- 
vision of a scale affects the original first division or 
the length of the scale. The adjustment is virtually 
equal to computing oblongs in a square form and serves 
a purpose of expediency through which a finite root 
may be extracted from any square in a most simple 
manner when it is desired to prove linear measures true 
by square contents. 

The rule for extracting adjusted roots is simply: To 
find the proper total of the given square by ratio between 
it and a standard square and to divide this total by the 
number of roots contained in the total of the nearest 
integral sq. (for it is evident that every sq. requiring 
adjustment of its root, must be a fractional sq.) By 
example : 

1. The contents of a described square is 5 76, the 
number of roots or rows of integrants contained in that 
sq. is 24. The contents shall be adjusted numerically 
to conform to the ratio common to described and in- 
scribed sq. viz: 2 : 1 and to a standard inscribed sq. 



2 2 MATHEMATICAL C0MMENSURAT10N. 

containing 289 integrants. What then shall be the 
adjusted root of the described sq.? 

Inscribed standard sq. 289. Ratio 1 : 2. Contents 
sought 578, because 1:21:289:578. 

Described sq. 578, number of roots in nearest integral 
sq. 24; 578^-24=24^ the number of the adjusted root. 

2. Inscribed sq. 288-^17 = 16^, adjusted root of in- 
scribed sq. 

Described sq. 576-5-2-^288 the adjusted contents of in- 
scribed sq. 

By this process a distinction is made by integral 
squares or squares where the areal unit-measures and 
the linear unit-measures agree and fractional squares or 
squares where the areal unit-measure and linear unit- 
measures disagree. In the former case the length and 
the root of a square's side are equal to each other and 
one. In the latter case, the number of roots in the square 
are equal to the linear unit-measures of the side ; and, 
the magnitude of the root multiplied by the number of 
roots gives the areal measure of the square. Or in other 
words, the linear measure multiplied by the root gives 
the true contents of a square. 

With these passing remarks intending to throw light 
upon the apparent incommensurable relationship of 
long and square measure which has puzzled all mathe- 
maticians for ages and which justly may be termed the 
idiosyncracies of mathematics, we will next proceed by 
an understanding of this adjustment to find the true 
relationship between an inscribed, a described, and an 
equal square. 



PREFATORY ESSAY. 

Diagram 3. 



23 




2424i9 x 7 24^X24=578. 

19 xi9=3 6l » 

17 Xi7=289- 

But, in oider to adjust the linear measurement of the 

circumference to the areal measures of the circle, let 

us suppose a circumference is given in the number 

912. 

Then the side of the equal square is 228. 
The circumference 9 I2-^3 2 4 8 1 9 gives 290JI7-5-1 7= 1 y 2 g for 
the diameter ; that is to say: the original circumference 
908 has by decrease in its unit-measures to the extent of 
§28 each, been increased in numerical value to the extent 
of 912 ; and the original diameter 289 has by propor- 
tionate decrease in its unit-measures been augmented in 
numerical value to the extent of 290227. In conse- 



24 MATHEMATICAL COMMENSURATION. 

quence of this radical change, the diagonal of the equal 
square has become even 19, instead of 18^ or i8||f and 
by this mutual adjustment an even ratio is obtained 
oetween chord and arc of a quadrant or of the sides of 
an inscribed and an equal square. This ratio is 17:19 
or 1:1$ . 

The diagram 3 shows that : 

The side of inscribed square is 1 7*2*7=1 7 f square con- 
tents : 290^7 = 289. 

The side of described square is 24^ =24. Square con- 
tents : 578=576. 

The side of equal square is 18^=19. Square con- 
tents; 359il=36i. 

The diagonal of inscribed square is 408 in final mul- 
tiples. 

The diagonal of the described square is 816 in final 
multiples. 

The diagonal of the equal square is 456 in final mul- 
tiples. 

Now, taking up the proposition of Legendre ; "that 
the area of a circle is equal to the area of a right-angled 
triangle, the altitude of which is equal to the radius, 
and the base equal to the circumference of the circle," 
we find ; 



PREFATORY ESSAY. 



25 



228 




.912 



Ratio: 289:908. 

By the ratio 289 : 908 :: 290^- : 912. 

We find the equal square's side to be 228. 

We find the square contents of the circle to be: 228X 
228=51984 final multiples which divided by the 361 
integrants of the equal square, as given in diagram 3, 
leave a final distribution of the multiples, which show 
each of the 361 integrants to contain 144 multiples, and 
consequently a commensurational unit-measure of & 
which applies alike to the root of the square, to the 
length of the side of the square, and to the number of 
roots in the equal square, and this last number (12) by 
further trial will prove to fit all the adjusted ratios given. 

RATIOS. 

17:24 between side and diagonal of square. 

289:908 between diameter and circumference of cir- 
cle. 

17:19 between chord and arc of quadrant. 

To one who has leisure and will take the time to 
carefully examine the matter, these demonstrations 
alone would be sufficient to convince the most sceptic; 



2 6 MATHEMATICAL COMMENSURATION. 

but to save this trouble we will revert back to the pro- 
position by Bernoulli and if we succeed in proving that 
these numbers correspond, and are commensurable to 
an algebraic formula which no one will deny to be 
true, a just claim has been presented for examination. 

Bernoulli's proposition. 

"If the number 4 be divided by i, 5, 9, 13, and 
every fourth number in succession, and afterwards by 
3, 7, n, 15, and every fourth number thereafter, the 
difference between the sum of the first set of quotients 
and that of the second is equal to the ratio between the 
diameter and the circumference." 

First Series: Second Series: 

585 "55 



4- i=4 4^ 3 = I 3 3 8 5 

4-*- 



5=o| 468 4-*- 7=07 660 

9=09 260 4-^11=0^ 420 

13=0* ISO 4-l5rr= 0l 4 5 308 



4 ^=1^ 
Sum of ig Sum of ig 

istsetofquo'ts5J?!J 2d set of quotients 2^ 
45°45 

5l 24871 

*8K 2 4*°2 

Difference: 3^ JJJ, 



PREFATORY ESSAY. 27 

In final multiples, this fraction, which has been call- 
ed the incommensurable ratio of diameter and circum- 
ference, is equal to 3x450454-769 = 135904. 

To show that these final multiples are commensurable 
to the final multiples of the circumferences: 912 and 
9o8,allweneeddo is to divide: 135904-5-91 2— 1495V 

For by making this fractional quotient into final 
multiples we obtain; 149x57+1=8494 which as a di- 
visor of 135904 divides the incommensurable(?)multiples 
without a remainder, because : 

8494) i359 4(i6 
8494 



50964 
50964 



This proves the ratio true, and Lambert's and Legen- 
dre's statement erroneous. 

For further proof that the ratio given by Bernoulli, 
and obtainable by algebra is commensurable to the ratio 
obtained by arithmetic, as shown in this treatise, let us 
demonstrate that ; 3J0I5 = x 5904-^908 =149237, and 149H! 
=33976 in final multiples will divide without a remain- 
der the first number : 

33976)i359 4(4. 

135904 



These figures prove conclusively that 908 and 912 are 
both commensurable to the number 135904, which is 



28 MATHEMATICAL COMMENSURATION. 

known to be the true expression for a commensurate cir- 
cumference, the relation of which is to the diameter 
as the relation of the two numbers 908: 289 or as diame- 
ter 1 is to circumference3 2 4 8 l 9 . 

Because of this commensurable relation, of both the 
numbers given, to the numbers obtained by algebraic 
formula, and because the algebraic ratio is not in itself 
cf a commensurable order for arithmetical computation, 
it is plain that the ratio 3^- or 135904 as a circumfer- 
ence is the mean between the linear and areal measure- 
ments of the circle's contents. Because if we take the 
mean of the two numbers : 33976 and 8494 we obtain : 

33976+8494=42470—2=21235 

And when we divide 135904 by 21235 we obtain; 

6|=32 which is the mean of the quotients 4 and 16, 
multiplied into area; 4X16=64-7-2=32. 

Presuming that sufficient evidence has been furnished 
to prove the subject worthy of further attention, 
and assuming that the ratios given are correct, let us 
next consider the advantages gained by the discovery 
and the possible effect it will have on the future meth- 
od of teaching geometry. 

It is worth to observe how the calculations obtained 
by the true ratio compare with calculations obtained 
by logarithms. 



PREFATORY ESSAY. 29 

When the side of a square is given as 1 7, the best 
logarithms approximate the diagonal by the number 
24.04, or 24^ whilst the true ratio gives 24 for the 
linear measure and 24^ for the areal measure of the 
root. 

In the square measure, logarithms give the number; 
57 6 looSo or 577iooooo whilst the true ratio gives the num- 
ber : 578. 

When the diameter is 10, logarithms give for the 
circumference the number 31.41592+ +etc. infinitely — 

By the true ratio : 31389 or 9080 in final multiples. 

By logarithms; diameter 289 gives circumference: 
907.920088. 

By the true ratio; diameter 289 gives circumference: 
908. 

By logarithms : diameter 10,000 gives circumference : 
3141592 etc. 

By the true ratio : diameter 10,000 gives circumfer- 
ence : 31418HI. 

These figures speak for themselves. 

At the diameter 10,000 we discover already a discre- 
pancy of two whole units from the true measure. 

What then will be the possible error in our estimate 
of the sun's distance from our planet with a supposed 
orbit of about 289 millions of miles? 

As to direct advantages gained by the use of 
this new method of computation, let us think of 



30 MATHEMATICAL COMMENSURATION, 

the time and labor it will save both scholar and tutor 
during a course of education in the mathematical 
branches. Think of constantly depending on volumes 
full of approximate logarithms every time we are to 
compute a geometric quantity when a few pages con- 
taining exact ratios and fixed rules giving correct re- 
suits will fill all requirement. Or, if from habit we 
must have logarithms let us have these constructed on a 
level foundation so that, instead of wading through 
millions to find a few numbers, we can find millions 
by manipulating a few units. 

Aside, however, from these considerations we ought 
to consider the narrowing effect all ready-made learning 
lias upon a pupil's mind during a course of education. 
The constant use of logarithmic tables in school, and 
the pupil's dependence upon these as text-book author- 
ity, lower the standard of geometry as a science and 
blunts the intellect by forcing the mind to accept 
what reason fails to show. Besides, scientific tuition 
without difinite explanation and clear understanding, 
sets aside a most important part of education, that of 
training the mind for rational thinking. 

Indirectly-^-the full effect of so radical a change in 
the mode of teaching geometry can only be vaguely 
conjectured; jet, every radical change effected through 
scientific discovery lead mostly to broad results how- 
ever simple and insignificant the first mention may 
appear. 



PREFATORY ESSAY. 3 1 

When Archimedes suggested that, had he a lever long 
enough and a fulcrum outside the earth, he could move 
the planet, it, at the time, was not understood, and 
little use was found for its application. Still, to invent- 
ive genius it was of great value and inaugurated as it 
were a revolution in mechanical arts. 

Galileo's first presentation of the moving planet was 
the inaugural step to an intellectual revolution in which 
scientific knowledge took up arms against superstitious 
belief. 

It is not impossible that this first presentation of 
commensuration applied to the most profound of 
sciences will lead to intellectual revolution in other 
directions and set people thinking over a possible 
equitable relationship in human affairs by applying the 
principles of commensuration to just debate on moral, 
social, civil and political reform. 



A complete series of five analytical primers ex- 
pressly prepared for students to the science of "Mathe- 
matical commensuration" and complete with examples 
and illustrations fully demonstrating the principles, is 
now prepared for publication, for copies of which the 
bublisher respectfully solicits your subscription. 



